RENEWAL THEORY FOR EMBEDDED REGENERATIVE SETS. BY JEAN BERTOIN Universite Pierre et Marie Curie

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1 The Aals of Probabiliy 999, Vol 27, No 3, RENEWAL TEORY FOR EMBEDDED REGENERATIVE SETS BY JEAN BERTOIN Uiversie Pierre e Marie Curie We cosider he age processes A A associaed o a moooe sequece R R of regeeraive ses We obai limi heorems i disribuio for A,, A ad for A,, A, which correspod o mulivariae versios of he reewal heorem ad of he DyiLamperi heorem, respecively Dirichle disribuios play a ey role i he laer Iroducio The rage R of a reewal process e, of a icreasig radom wal forms a discree regeeraive se which ca be viewed as he se of reewal epochs of some recurre eve i he sese of Feller 4, 5 From he poi of view of he prese wor, reewal heory is cocered wih he asympoic behavior as ime goes o of he so-called age process, A ifs 0: s R 4 The mai resuls i his field are he reewal heorem, which is he ey o he limi heorem for A i he case whe he reewal process has a fiie mea, ad he DyiLamperi heorem, which reveals he role of geeralized arcsie laws i he sudy of he ormalized age A The purpose of his wor is o develop a mulivariae reewal heory for a moooe sequece of regeeraive ses, R R, where he embeddig is compaible wih he regeeraive propery he meaig of compaible will be made precise i he ex secio Specifically, we will give a explici formula for he joi Laplace rasform of he age processes A,, A ad prese a mulivariae exesio of he reewal heorem ad of he DyiLamperi heorem I he laer, he geeralized arcsie ie, bea law which appears as a limi i he sadard siuaio is replaced by a Dirichle disribuio i he mulivariae versio A oiceable feaure i our resuls is ha hey deped oly o he idividual disribuios of he regeeraive ses R,, R ad o o heir joi disribuio as oe migh have expeced I paricular, hey do o seem o be direcly relaed o he mulidimesioal versio of he reewal heorem i 8, as he laer requires he owledge of he joi dyamic of he age processes For he sae of compleeess, we refer he reader o 7 ad he refereces herei for mulidimesioal exesios of he reewal heorem i a differe vei Received Jue 998 AMS 99 subjec classificaio Primary 60K05 Key words ad phrases Mulivariae reewal heory, regeeraive se, Dirichle disribuio 523

2 524 J BERTOIN The ex secio coais he precise saemes ad proofs i his discree seig Secio 3 deals wih he exesio o coiuous ie, odiscree regeeraive ses 2 Discree seig 2 Prelimiary We firs recall some basic feaures i regeeraive ses ad iroduce releva oaio For he sae of simpliciy, we implicily oly discuss ubouded regeeraive ses, as bouded regeeraive ses are o releva o he scope of his wor A closed ubouded radom se R 0, is called a regeeraive se if i fulfills he regeeraive propery Tha is, if F 0 deoes he filraio iduced by he characerisic fucio, he for every F R -soppig ime T such ha T R as, he righ-had porio of R as viewed from T, RT s 0: s T R 4, is idepede of FT ad has he same law as R Whe he regeeraive se R is discree, which we assume from ow o hroughou his secio, here exiss a uique reewal process S S, S, 0 such ha R S : 4 More precisely, S where he iid radom variables, correspod o he waiig imes bewee successive occurreces of R for he sae of simpliciy, we shall wrie for a radom variable disribued as i The disribuio of R is he characerized by he fucio q exp qs e q, q 0 The reewal measure associaed wih R is give by ž Ý B4/ Ý R 0 U B S B, B B 0, ; is Laplace rasform is simply The age process qx 0 e U dx, q 0 q A ifs 0: s R 4, 0 is a srog Marov process which vaishes exacly o R Is oe-dimesioal disribuio is give by A dx x U dx, 0 x Whe he reewal process is olaice e, here is o r 0 such ha R r wih probabiliy ad has fiie mea m, he sadard reewal heorem combied wih yields ha A coverges i law as owards he probabiliy measure m x dx I he ifiie mea case,, here is a limi heorem for he ormalized age A provided ha

3 EMBEDDED REGENERATIVE SETS 525 he sep disribuio belogs o he domai of aracio of some sable law More precisely, Tauberia heorems ad are he ey o he DyiLamperi heorem, which roughly saes ha A coverges i disribuio if ad oly if is mea coverges We sress ha he reewal heorem ad he DyiLamperi heorem are he oly odegeerae limi heorems for he age process; i paricular, oe would ge ohig ieresig by ormalizig he age process by oher powers of Now cosider a sequece of 2 embedded regeeraive ses, 2 R R Noaio usig superscrips such as should be clear from he coex For every i,,, we iroduce F 0, he filraio geeraed by he characerisic fucios,, R R, afer he usual compleios DEFINITION We say ha he embeddig 2 is compaible wih he rege- eraive propery if for every i,, ad every F -soppig ime T wih T R as, he shifed ses j j R T s 0: s T R 4, j i,, are joily idepede of F ad have joily he same law as R,, R T This oio has bee iroduced for 2 i he more geeral coiuous seig i where i is called regeeraive embeddig Le us prese a few examples i which 2 holds ad is compaible wih he regeeraive propery, ad refer o for more Cosider a moooe sequece L L of sublaices of d ad a d radom wal W W, valued i The ae 4 R : W L 2 Suppose ha S,, S are idepede ieger valued reewal processes, ad cosider he compoud processes S S S, i,, The each S is a reewal process his is a discree versio of Bocher s subordiaio ad he we ca ae R S, 4 3 Suppose ha R,, R are idepede regeeraive ses, ad se R R R for i,, 4 Cosider W,, W, a-dimesioal radom wal such ha for j,,, he real-valued radom wal W j W j icreases ie, is a reewal process The ae for R he se of ascedig ladder epochs of W, ha is, he se of imes whe W reaches a ew maximum More geerally, give age processes A,, A, 2 holds if A A, ad is compaible wih he regeeraive propery if he i -uple A,, A is Marovia for each i,, We poi ou ha he codiio ha he age processes are joily Marovia appears i 8 O he oher had, i is easy o cosruc examples where 2 holds ad is compa-

4 526 J BERTOIN ible wih he regeeraive propery hough he age processes are o joily Marovia See, for isace, he couer-example afer he proof of Lemma We assume from ow o ha he embeddig 2 is compaible wih he regeeraive propery I is clear ha his assumpio imposes some resricio o he fucios,,, ad more precisely, oe ca sae he followig lemma LEMMA For every i,,, he raios are compleely moooe fucios We sress ha, coversely, if he raio 2,, are compleely moooe, he here exiss regeeraive ses R,, R whose idividual disribuios are characerized by he fucios,,, respecively, such ha 2 holds ad is compaible wih he regeeraive propery As we shall o eed his feaure i he sequel, we jus refer o for a complee argume PROOF OF LEMMA This is esseially he direc par of Theorem i ; for he sae of compleeess we prese here he proof i he discree seig We may suppose ha 2 Fix a arbirary q 0 ad recall ha qx 2 q e U dx ž Ý / e 2 q 0, 2 R We iroduce he pariio geeraed by R, so ha he righ-had side ca be expressed as ž / Ý Ý S S 4exp qs exp q S 0 R 2 We he apply he regeeraive propery a S o ge ž Ý S S 4 2 / R e q exp qs ž Ý S S 4 / exp qs exp q S R 2 S ž / Ý S 4 exp qs e q 2 R Taig he sum over, we fially ge q 2 ž Ý S 4 e q q / 2 2 The raio is hus he Laplace rasform of some measure o 0,, which proves our claim R

5 EMBEDDED REGENERATIVE SETS 527 A his poi, i is also impora o sress ha he fucios,, do o deermie he joi law of R,, R ; le us prese a simple couerexample Suppose ha he law of he geeric sep 2 of S 2 is 2 2 a b 2 for some a b Nex, cosider he smalles eve ieger 2 such ha 2 2 a ad 2 2 b, ad pu S S2 2 Ieraig he cosrucio i a obvious ie, regeeraive way, we obai a reewal process S wih rage R R 2 where he embeddig is compaible wih he regeeraive propery Plaily, he ex o las ad he las waiig imes bewee occurrece imes of R 2 before a occurrece of R have respecive duraios a ad b The do he same cosrucio afer exchagig he role of a ad b We ge aoher regeeraive se R R 2, where he embeddig is compaible wih he regeeraive propery, bu ow he las-bu-oe ad he las waiig imes bewee occurrece imes of R 2 before a occurrece of R have respecive duraios b ad a The joi 2 2 disribuios of R, R ad of R, R hus differ Noeheless, i is obvious ha he variables S ad S have he same law, so he fucios ad are ideical Uforuaely, i seems here is o simple aalogue of he ideiy for he joi oe-dimesioal disribuio for he age processes A,, A owever, we poi ou ha some joi Laplace rasform has a simple expressio, which is he ey echical poi of his sudy For every q,,, 0, we have LEMMA 2 q 2 ž 0 d e exp A A A A A 4/ q 2 q 2 q 2 q q q q PROOF We firs suppose ha 2 ad deoe he lef-had side i he formula of he saeme by F,, q We decompose he iegral over 0, accordig o he regeeraive se R S, S, 4 0 ; ha is, we wrie q S Ý 0 0 S d e exp qs d exp q S I he sum, we apply he regeeraive propery a S As A,, A all vaish a each S ad as A grows liearly o he ierval S, S, we ge F,, q S ž Ý q 2 / ž 0 0 4/ exp qs d e exp A A

6 528 J BERTOIN The firs erm i he produc equals q The observe ha he secod erm i he produc is ideical o he oe which appears i he same formula for F 0,,, Specifically, he same calculaio gives q 2 F 0,,, q 2 exp q S ž Ý / 0 S ž 0 We deduce ha d exp q 4/ exp A 2 A 3 A 2 / S q 2 ž d e exp A A q 0 q Fq,, Fq 0, 2,, q Ieraig he argume, we obai F,, q 2 q 2 q q q 2 q Fq 0,,0 q This is he desired resul, as obviously F 0,,0 q q Fially, he geeral case whe he sequece,, is o ecessarily moooe follows by aalyic coiuaio REMARK There is o similar formula for he joi Laplace rasform of residual lifeimes R if s 0: s R 4 The reaso is ha heir joi disribuio depeds o he joi law of he regeeraive ses R,, R ad o merely o he idividual laws 22 A mulivariae reewal heorem We firs develop some maerial o sae he mulivariae versio of reewal heorem Recall from Lemma ha he raios 2,, are compleely moooe, ad iroduce he correspodig measures,, o 0, ; ha is, qx i 0, q e dx, q 0 q

7 EMBEDDED REGENERATIVE SETS 529 For i, we defie by or equivalely, e qx dx q q, q 0, 0, dx S x dx, x 0, We also iroduce he iverse of he mass q ci lim, i,,, 0, q0 q i where we agree ha q q Observe ha q c c lim, q S q0 so all he coefficies c,, c are posiive if ad oly if S, ad he c,, c are probabiliy measures We ow sae he mulivariae reewal heorem TEOREM Suppose ha he reewal process S is olaice ad has fiie mea The he -uple A A 2,, A A, A coverges i disribuio as owards he produc measure c c PROOF Le T be a idepede expoeial ime wih parameer ad q 0 Accordig o Lemma 2, he joi Laplace rasform of AT q 2 A,, A A, A evaluaed a,, is T q T q T q T q q q q 2 q q q Whe q 0, his quaiy coverges owards he Laplace rasform of c c Thus, all ha we eed is o chec he covergece i disribuio of he 2 -uple A A,, A A, A, as we have ideified he oly possible limi To his ed, ae real umbers x,, x 0 ad cosider A A 2 x,, A A x, A x Ý S 4 S S S 4 A A 2 x,, A ž x 4 0 /

8 530 J BERTOIN Jus as i he proof of Lemma 2, we apply he regeeraive propery a S o rewrie his quaiy as where for s 0, ž Ý S 4 / 0 f S g S 2 f s As As x,, As As x, As x S s ad g s S s I oher words, if we se f 0o, 0, he we have f s g s U ds, 0, where U is he reewal measure of he reewal process S We ow see from ha f A The sadard reewal heorem eails ha A coverges i law as owards some absoluely coiuous disribuio Wriig he codiioal probabiliy f s as a quoie of ucodiioal probabiliies, i is easily show ha he fucio f is bouded ad has a mos couably may discoiuiies, so we see ha he desired covergece holds Of course, here also exiss a versio of Theorem i he laice case which we ow sae TEOREM 2 Suppose ha he reewal processes S,, S are laice wih ui spa i he sese ha he group geeraed by he suppor of he sep disribuio of S is for every i If S has fiie mea, he for all oegaive iegers a,, a, A A 2 a,, A A a, A a coverges as he ieger goes o ifiiy oward c a a, where q q Ý e for i,,, q ad 0 e q Ý 0 q q e c S lim q q q0 The argume for he proof follows he same lie as i he olaice case, usig geeraig fucios isead of Laplace rasforms We sip he deails

9 EMBEDDED REGENERATIVE SETS A mulivariae DyiLamperi heorem We ow ur our aeio o he asympoic behavior of he reormalized age processes A,, A By he sadard DyiLamperi heorem, A coverges i disribuio as if ad oly if he followig limi exiss: 3 lim A 0, The is regularly varyig a 0 wih idex i ad he limi disribuio of A is he so-called geeralized arcsie disribuio wih parameer ie, bea wih parameers, i i i See, for isace, Secio 862i3 Therefore, we may ad will assume from ow o ha 3 holds for i,, As he embeddig hypohesis R R implies ha A A, we mus have 0 I order o focus o he mos ieresig case, we shall assume ha hese iequaliies are sric, Ideed, if we had, for example,, he A A 2 would coverge i probabiliy o 0 ad we migh jus as well igore A 2 The mulidimesioal aalogues of he bea disribuios are he Dirichle disribuios More precisely, recall ha a -dimesioal radom variable X,, X has a Dirichle disribuio wih parameers,, 0, if X dx,, X dx i x x Ý xi dxi i for x,, x i he -dimesioal simplex ie, x i 0 ad x x We are ow able o sae he mulivariae versio of he DyiLamperi heorem TEOREM 3 Suppose ha 3 ad 4 hold The 2 ž A A,, A A, A / coverges i disribuio as owards a -dimesioal Dirichle disribuio wih parameers 2,,,, I order o prove he resul, i is more coveie o wor wih he odecreasig processes G A sup s : s R 4 i

10 532 J BERTOIN Firs, we ivesigae he limi behavior of some joi Laplace rasform of G,, G LEMMA 3 Fix q,,, 0 The q lim d e exp G G q q q q 0 PROOF Iroduce he degeerae regeeraive se R 04 which corre- 0 0 spods o A ad We deduce from Lemma 2 beware of he chages of idices ha q d e exp G G q q 2 q 0 q q q q of course, he firs erm i he produc simply equals Muliply q,,, by ad recall from he hypohesis 3, ha is regularly varyig a 0 wih idex i This yields our saeme Nex, we iver he mulivariae Laplace rasform which appears i Lemma 3 LEMMA 4 For every q,,, 0, we have 2 2 q q q q d e d exp q d e PROOF To sar wih, recall ha q d exp 0 exp q 2

11 This yields EMBEDDED REGENERATIVE SETS q 2 q d exp 0 2 s ds exp q s d exp 0 d exp exp q, ad we ierae he calculaio We are ow able o characerize he asympoic behavior i disribuio of G,, G COROLLARY Suppose ha 3 ad 4 hold The probabiliy measure G d,, G d, 4 o 0 coverges wealy as 0 oward d d 2 PROOF For each 0, he processes G are odecreasig, so he fucio exp G G ž 4/ is oicreasig We he obai from Lemmas 3 ad 4 ad by a iegraio by pars ha ž / lim eq d ž exp G G 4/ 0 0 q ž e d d exp / 0 2 d exp As his holds for all q 0, i implies ha for almos every 0, ž 4/ lim exp G G 0

12 534 J BERTOIN d exp 0 d exp, which readily eails our claim Theorem 3 follows ow from Corollary by a immediae chage of variables 3 Coiuous seig We ow ur our aeio o he coiuous seig; le us firs briefly prese basic oios i ha field ad refer o Frised 6 or 2 for deails The coiuous-ime aalogue of a reewal process is a so-called subordiaor ha is, a icreasig Levy process, which will be deoed geerically by 4, 0 The closed rage R, 0 cl of a subordiaor is a regeeraive se i he sese give a he begiig of Secio 2, ad coversely, ay discree or odiscree regeeraive se ca be viewed as he closed rage of some subordiaor The disribuio of is characerized by is Laplace expoe, exp q exp q 4,, q 0, ad he Laplace expoe is give by he LevyKhichie formula 0, q dq e qx dx ere d 0 is he drif coefficie, he Levy measure; loosely speaig, he Levy measure describes he disribuio of gaps i R ad ca be hough of as he aalogue of he sep disribuio of he reewal process he absece of illig erms is because we implicily focus o ubouded regeeraive ses The reewal measure U of is is poeial measure; ha is, Udx 0 qx dx d, ad is Laplace rasform is e U dx q 0 Plaily, he oio of compaibiliy wih he regeeraive propery for a sequece of embedded regeeraive ses also maes sese i he coiuous seig The he mulivariae versios of he reewal heorem ad of he DyiLamperi heorem sill hold rue The proofs follow esseially he same roue as i he discree seig Techically, Maisoeuve s exi sysem formula ad excursio heory provide he righ ools for adapig he argumes give i Secio 2 o he coiuous seig; we sip he deails he ieresed reader is referred o he proof of Theorem i for he coiuous versio of Lemma ; he modificaios which are eeded for he oher seps are i he same vei O he oher had, we poi ou ha he coiuous versio of Theorem 3 also holds for small imes, ha is, whe is replaced by 0 i he saeme Moreover, here is a versio a fixed ime for self-similar e, sable regeeraive ses

13 EMBEDDED REGENERATIVE SETS 535 PROPOSITION Suppose ha 4 holds ad ha q q i for every q 0 ad i,, The for every 0, 2 ž A A,, A A, A / has a -dimesioal Dirichle disribuio wih parameers 2,, REFERENCES BERTOIN, J 997 Regeeraive embeddig of Marov ses Probab Theory Relaed Fields BERTOIN, J 999 Subordiaors: Examples ad Applicaios Ecole d ee de Probabilies de S-Flour XXVII Lecure Noes i Mah Spriger, Berli To appear 3BINGAM, N, GOLDIE, C M ad TEUGELS, J L 987 Regular Variaio Cambridge Uiv Press 4FELLER, W E 968 A Iroducio o Probabiliy Theory ad Is Applicaios, 3rd ed Wiley, New Yor 5FELLER, W E 97 A Iroducio o Probabiliy Theory ad Is Applicaios, 2, 2d ed Wiley, New Yor 6FRISTEDT, B E 996 Iersecios ad limis of regeeraive ses I Radom Discree Srucures D Aldous ad R Pemale, eds 25 Spriger, Berli 7 OGLUND, T 988 A mulidimesioal reewal heorem Bull Sci Mah SPITZER, F 986 A mulidimesioal reewal heorem I Probabiliy, Saisical Mechaics, ad Number Theory G C Roa, ed 4755 Academic Press, Orlado LABORATOIRE DE PROBABILITES, UMR 7599 UNIVERSITE PIERRE ET MARIE CURIE PARIS FRANCE F jbe@ccrjussieufr